Groupes de Picard et problèmes de Skolem. I. (Picard groups and Skolem problems. I). (French) Zbl 0704.14014

A problem of Skolem asks about the existence of a solution of a system of diophantine equations over a ring R of algebraic integers whose coordinates belong to a finite extension of R. In the scheme-theoretical language this is a question about the existence of a multi-section of a morphism of finite type \(f:X\to Spec(R)\), where R is a Dedekind ring.
The main result of this paper asserts that the answer is positive if the following conditions are satisfied:
(i) R is an excellent ring whose residue fields are algebraic over a finite field;
(ii) for any finite extension \(K'\) of the field of fractions K of R the normalization \(R'\) of R in \(K'\) has torsion Picard group;
(iii) there exists \(K'\) as above such that one of the irreducible components of the base change \(X\otimes_ RR'\) is mapped surjectively to \(Spec(R').\)
This result is a generalization of a theorem of R. S. Rumely [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)], who assumed that R is a ring of integers in an algebraic number field and f is surjective. The proof is geometric and does not use the theory of capacity of Rumely.
[See also the following review.]
Reviewer: I.V.Dolgachev


14G25 Global ground fields in algebraic geometry
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
11R04 Algebraic numbers; rings of algebraic integers
11D41 Higher degree equations; Fermat’s equation
Full Text: DOI Numdam EuDML


[1] M. ARTIN , Some Numerical Criteria for Contractibility of Curves on Algebraic Surfaces (Amer. J. Math., vol. 84, 1962 , p. 485-496). MR 26 #3704 | Zbl 0105.14404 · Zbl 0105.14404
[2] D. CANTOR et P. ROQUETTE , On Diaphantine Equations over the Ring of All Algebraic Integers (J. of Number Theory, vol. 18, 1984 , p. 1-26). MR 85j:11036 | Zbl 0538.12014 · Zbl 0538.12014
[3] P. DELIGNE et D. MUMFORD , The Irreducibility of the Space of Curves of Given Genus (Pub. Math. I.H.E.S., n^\circ 36). Numdam | Zbl 0181.48803 · Zbl 0181.48803
[4] W. FULTON , Intersection Theory , Springer Ergebnisse, vol. 2. · Zbl 0541.14005
[5] J.-P. JOUANOLOU , Théorèmes de Bertini et applications (Progress in Math., vol. 42, Birkhöuser). MR 86b:13007 | Zbl 0519.14002 · Zbl 0519.14002
[6] D. KNUTSON , Algebraic Spaces (Lecture Notes in Math., vol. 203, Springer). MR 46 #1791 | Zbl 0221.14001 · Zbl 0221.14001
[7] L. MORET-BAILLY , Points entiers des variétés arithmétiques [Séminaire de théorie des nombres de Paris (Progress in Math., Birkhöuser)]. Zbl 0644.14008 · Zbl 0644.14008
[8] L. MORET-BAILLY , Groupes de Picard et problèmes de Skolem II [Ann. sci. Ec. Norm. Sup. (4e série, t. 22, 1989 , p. 181-194)]. Numdam | MR 90i:11065 | Zbl 0704.14015 · Zbl 0704.14015
[9] L. MORET-BAILLY , Problèmes de Skolem sur les champs algébriques (en préparation). · Zbl 1106.11022
[10] D. MUMFORD , The Topology of Normal Singularities of an Algebraic Surface and a Criterion for Simplicity (Pub. Math. I.H.E.S., n^\circ 9, 1961 ). Numdam | MR 27 #3643 | Zbl 0108.16801 · Zbl 0108.16801
[11] R. RUMELY , Capacity Theory on Algebraic Curves (à paraître). · Zbl 0679.14012
[12] R. RUMELY , Arithmetic Over the Ring of all Algebraic Integers (J. reine u. angew. Math., 368, 1986 , p. 127-133). MR 87i:11041 | Zbl 0581.14014 · Zbl 0581.14014
[13] T. SKOLEM , Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten (Skrifter Norske Videnskaps-Akademi i Oslo, Mat. Naturv. Kl. 10, 1934 ). Zbl 0011.19701 | JFM 61.0175.06 · Zbl 0011.19701
[14] L. SZPIRO et coll. Séminaire sur les pinceaux de courbes de genre au moins deux (Astérisque, vol. 86). · Zbl 0463.00009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.